## Abstract

A maximal partial Hamming packing of Z^{n}_{2} is a family S of mutually disjoint translates of Hamming codes of length n, such that any translate of any Hamming code of length n intersects at least one of the translates of Hamming codes in S . The number of translates of Hamming codes in S is the packing number, and a partial Hamming packing is strictly partial if the family S does not constitute a partition of S. A simple and useful condition describing when two translates of Hamming codes are disjoint or not disjoint is proved. This condition depends on the dual codes of the corresponding Hamming codes. Partly, by using this condition, it is shown that the packing number p, for any maximal strictly partial Hamming packing of Z^{n}_{2}, n = 2 ^{m} -1, satisfies m+1 ≤ p ≤ n-1 . It is also proved that for any n equal to 2 ^{m} -1, m ≥ 4 , there exist maximal strictly partial Hamming packings of Z^{n}_{2} with packing numbers n-10,n-9,n-8,...,n-1. This implies that the upper bound is tight for any n = 2 ^{m} -1, m ≥ 4 . All packing numbers for maximal strictly partial Hamming packings of Z^{n}_{2} , n = 7 and 15, are found by a computer search. In the case n = 7 the packing number is 5, and in the case n = 15 the possible packing numbers are 5,6,7,...,13 and 14.

Original language | English |
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Pages (from-to) | 335-355 |

Number of pages | 21 |

Journal | DESIGNS CODES AND CRYPTOGRAPHY |

Volume | 42 |

Issue number | 3 |

DOIs | |

Publication status | Published - Mar 2007 |

MoE publication type | A1 Journal article-refereed |

## Keywords

- Perfect codes